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Complex Numbers- Introductory Tutorial, Examples, Solved Problems, MCQ Quiz - Argand Plane, Roots of Unity, Basic Operations


 



 
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Here's an outline and a summary of what's introduced in this tutorial

Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE 

Main or Advanced/AIEEE

, and anyone else who needs this Tutorial as a reference!


After studying the topic you might also be interested in attempting the following 2 Quizzes related to Complex Numbers.


Complex Numbers

Basics of Complex Numbers, Real and Imaginary Parts, Iota

  • Complex Numbers is the largest and the complete set of numbers, consisting of both real and unreal numbers.
  • A complex number is of the form
    a+ib, where a and b are both real numbers, and i is called “iota” and is defined to be the square root of -1. That is, i=√-1. Also, i
    2=-1.
  • A complex number is usually denoted by the letter ‘z’.
  • ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number.
  • The notion of complex numbers increased the solutions to a lot of problems. For instance, had complex numbers been not there, the equation x2+x+1=0 had had no solutions.

    Because of ‘i', we can now extend the square root to be defined for negative numbers as well. If y is a negative number, then y can be written as -|y| where |y| is the absolute value of y. Then, the square root of y is defined to be equal to be

    y=i√|y|


Algebra of Complex Numbers, Conjugate, Modulus and Argument, Euler's Form

  • Complex Numbers have different algebraic laws as well. Let z1=a+ib,z2=c+id.
    • z1+z2=(a+c)+i(b+d)
    • z1-z2=(a-c)+i(b-d)
    • z1.z2=(ac-bd)+i(ad+bc)
    • z1/z2=((ac+bd)+i(ad-bc))/(c2-d2)
    • 1/z1=(a-ib)/(a2-b2)
  • Conjugate : Conjugate of a complex number z=a+ib is defined as

z=a-ib.


  • Modulus of z is written as |z| and is defined to be equal to √(a2+b2)
  • Thus, z.Conjugate(z)=|z|2.
  • Commutative, Associative, and Distributive Laws of algebra apply to Complex Algebra as well.
  • Complex Numbers can be represented pictorially on a plane known as Argand Plane.
  • The plane consists of a Real Axis, and an Imaginary Axis.
  • The angle that z makes with the real axis (counterclockwise) is called the argument of z, and the length of line segment joining z and the origin is equal to |z|, also sometimes written as ‘r’.
  • Different representations of z,
    z=a+ib
    z=r(cosθ + i sinθ) [Modulus Argument Representation]
    z=re
    [Euler’s Form]
  • Observe that, e=r(cosθ + i sinθ)
  • Multiplication and Division using using Euler’s Form:
    Let z
    1=r1eiθ1, z2=r2eiθ2
    z
    1z2=r1r2ei(θ1+θ2)

z1/z2=(r1/r2)ei(θ1-θ2)


  • Conversion between various forms.
    r=|z|;
    Let α=tan
    -1(|b/a|); then θ=α,π-α,α-π,-θ depending upon z being in first, second, third or fourth quadrant respectively. This is called the principal argument.
  • Approaches to solving questions:
    • Expand every complex number as a real and imaginary part.
    • Use Euler’s form for every Complex Number
    • Use Algebraic Identities, and solve in general, keeping in mind that complex numbers have a real and an imaginary part.


Complex Numbers as Free Vectors

  • Another approach to dealing with Complex Numbers is to treat them like free vectors. Free Vectors are vectors that can be moved around without changing direction, in other words, that do not really matter upon the origin. The approach that was used so far was treating complex number as a point.
  • Any comple number such as 2+3i can be any one of the infinite number of vectors
  • e is multiplied with the vector which is to be rotated, and rotates the vector by “θ” anticlockwise. Clockwise rotation is obtained by multiplying with e-iθ
  • m, n are unit vectors; and we have ne=m, m e-iθ=n

Here's a quick look at some of the questions we will solve

  • What is the value of (1+i)(1+i2)(1+i3)(1+i4)?
  • If (1+i)(1+2i)(1+3i). . .(1+ni)=a+ib, then what is  2X5X10X…X(1+n2)?
  • If (x+iy)1/3=a+ib; then x/a+y/b=?
  • Find the sum of i+i2+i3+…upto 1000 terms.
  • If |z|=2, find the area of the triangle formed by z,iz,z+iz.
  • If x=-5+2√-4, find the value of x4+9x3+35x2-x+4.
  • Find the square root of 7-24i.
  • Given that cosα+cosβ+cosγ=0 and sinα+sinβ+sinγ=0, prove that  sin3α+sin3β+sin3γ=3sin(α+β+γ)
  • If cosα+2cosβ+3cosγ=0 and sinα+2sinβ+3sinγ=0. Show cos3α+8cos3β+27cos3γ=3sin(α+β+γ).
  • Given that 1,α1,α2,…,αn-1 are nth roots of unity, Show that (1- α1)(1- α2)…(1- αn-1)=n.
  • Show that sin(π/n)sin(2π/n)…sin((n-1)π/n)=n/2n-1

Complete Tutorial Document with Examples and Solved Problems (There is an MCQ Quiz after this)




MCQ Quiz #1

Companion MCQ Quiz/Worksheet #1 for Complex Numbers (The Basics) - test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz: Complex Numbers- The Basics



MCQ Quiz/Worksheet #2: More on Complex Numbers

Read the questions in the document below, and fill up the answers in the answer submission form below it.
Your score will be emailed to you

Questions: More on Complex Numbers


Answer Submission Form for MCQ Quiz #2

Answer Submission Form for MCQ Quiz #2: More on Complex Numbers






You might like to take a look at our other algebra tutorials:

 Introduction to Complex Numbers
Introduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. N-th roots of a complex number. Notes, formulas and solved problems related to these sub-topics.
 The Principle of Mathematical Induction Introductory problems related to Mathematical Induction.Quadratic Equations
Introducing various techniques by which quadratic equations can be solved - factorization, direct formula. Relationship between roots of a quadratic equation.  Cubic and higher order equations - relationship between roots and coefficients for these. Graphs and plots of quadratic equations.
Quadratic Inequalities
 Quadratic inequalities. Using factorization and visualization based methods.
 Series and Progressions
Arithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them.